System and Method for the Generation of Multiple Angle Correlation-Based Digital Watermarks

ABSTRACT

Disclosed are systems and methods directed to the generation of digitally watermarked grayscale images having a watermark embedded at different arbitrary Rotation Angles. A single public key can be used to retrieve the otherwise invisible watermarks from the watermarked images when overlaid atop the images at Orientation Angles corresponding to the Rotation Angles.

CROSS-REFERENCE TO COPENDING APPLICATIONS

Attention is directed to co-pending applications filed concurrentlyherewith: U.S. application Ser. No. ______, Attorney Docket No.20051772-US-NP, entitled “SYSTEM AND METHOD FOR THE GENERATION OFCORRELATION-BASED DIGITAL WATERMARKS”; and U.S. application Ser. No.______, Attorney Docket No. 20060585-US-NP, entitled “SYSTEM AND METHODFOR THE GENERATION OF MULTI-LAYER CORRELATION-BASED DIGITAL WATERMARKS”the disclosure found in these co-pending applications are herebyincorporated herein by reference in its entirety. The appropriatecomponents and processes of the above co-pending applications may beselected for the teaching and support of the present application inembodiments thereof.

BACKGROUND AND SUMMARY

Disclosed in embodiments herein are methods and systems for generationof separate gray-scale watermarked images having watermarks that can beretrieved, or viewed, using the same public key rotated to differentorientation angles.

The basics of phase-shift based digital watermarks, orcorrelation-marks, are described in U.S. Pat. No. 6,252,971 for “Digitalwatermarking using phase-shift stoclustic screens,” by S. Wang,previously incorporated herein by reference. Briefly, if two similarcluster halftone patterns are superimposed on each other, the outputappearances can differ significantly depending on the relativepositions, or the phase shift, of the two patterns. For example, the twocheckerboard patterns, 100 depicted in FIGS. 1A and 110 depicted in FIG.1B, are essentially the same, except that the pattern 110 in FIG. 1B isa shifted version of the pattern 110 in FIG. 1A with an exactly “one-boxwidth” shift. If the two patterns 100 and 110 are superimposed on eachother with a perfect alignment, the result would be a completely blackimage as depicted by 120 in FIG. 1D. On the other hand, overlappingpattern 100 of FIG. 1A with itself, which can be considered anotherversion of 100 with a zero-shift, gives an identical pattern to theoriginal pattern 100, as depicted by 130 in FIG. 1C.

Referring now to FIGS. 2A-2C, a halftone pattern is shown at 200 havingonly it's central portion, shown as region 210, shifted in this manner.When the reference, or “public key”, represented by the halftone pattern240 of FIG. 2B is overlaid on top of the pattern 200, the result isclearly visible as a black central region shown at 250 in FIG. 2C. Theexample depicted in FIGS. 2A-2C is a simple demonstration for thephase-shift digital watermark technique. The shifted central part 210 inthe picture may be considered as a square watermark, which is retrievedas a black square 250 in the overlay shown in FIG. 2C. The shiftrequired for an optimal retrieval is equal to a half period of thehalftone structure, or π, in a general mathematic term. The problem witha simple “insertion”, however, is that the boundaries between theshifted portion 210 and the balance of the image are quite visible as aseam 220 shown in FIG. 2A. To hide the seam 220, the phase jump fromzero to π should be replaced by a smooth phase transition.

Prior patents, such as U.S. Pat. No. 6,252,971 for “Digital watermarkingusing phase-shift stoclustic screens,” by S. Wang, hereby incorporatedherein by reference in its entirety, describe a method to embedcorrelation-based phase-shift digital watermarks, also referred to acorrelation marks, into halftone screens. By overlaying a transparencyon the prints generated by the special halftone screen, for example as apublic key, an invisible watermark embedded in the image can beretrieved.

An aspect of the disclosed system and method provided extends use andformation of correlation marks by enabling separate watermarked imagesto be formed by embedding a phase shifted watermark into an image byhalftoning the image with the three-dimensional threshold array where atleast one input thereto is a phase shift value using a halftonestructure having a plurality of different, arbitrary Rotation Angles.The same public key, in the form of a transparency, can be used toretrieve the watermark from all of the images when overlaid atop eachimage at an Orientation Angle matching the Rotation Angle used forembedding the watermark in the image.

Disclosed in embodiments herein is a method for digital watermarking ofa grayscale image, including receiving the grayscale image to bewatermarked, determining the grayscale watermark to be embedded in theimage, and creating a plurality of digital watermarked grayscale imagesby embedding a digital watermark into the image by halftoning the imagewith a three-dimensional threshold array where at least one inputthereto is a phase shift value using a halftone structure defined by thespatial vectors V_(a)(x_(a), y_(a)) and V_(b)(x_(b), y_(b)) having thesame interior angle α (the angle between the two vectors V_(a) andV_(b)), the same vector amplitude D, and a different Rotation Angle foreach.

Disclosed in embodiments herein is a method for retrieving a watermarkimage from a plurality of digitally watermarked grayscale images, eachformed by halftoning an image with a halftone structure having adifferent arbitrary Rotation Angle using a three-dimensional thresholdarray having phase shift value as an input, including overlaying asingle public key transparency of a checkerboard pattern having halftonefrequencies and angles matching the halftone structure used for formingthe plurality of digital watermarked grayscale images atop each of thewatermarked images and orienting the key with respect to the watermarkedimages at Orientation Angles matching the Rotation Angles used forembedding the watermark image into the corresponding digitallywatermarked image to retrieve the watermark image.

Also disclosed in embodiments herein is a system for producing digitalwatermarked images including an input image source, image memory forstoring the input image to be watermarked, watermark memory for storingthe watermark to be embedded in the image, and an image processorincluding a three-dimensional threshold array where at least one inputthereto is a phase shift value for embedding an invisible digitalwatermark into the image by halftoning the image with thethree-dimensional threshold array using a plurality of halftonestructures each defined by the spatial vectors V_(a)(x_(a), y_(a)) andV_(b)(x_(b), y_(b)) having the same interior angle α, the same vectoramplitude D, and a different Rotation Angle to produce a plurality ofdigitally watermarked output images.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B are exemplary representations of halftone patterns andFIGS. 1C and 1D illustrate the effect achieved by overlaying thepatterns of FIGS. 1A and 1B;

FIGS. 2A-2C are exemplary representations of an aspect of embodiments inthe present invention showing the phase shifting of only a portion of ahalftone image;

FIGS. 3A and 3B are representative examples of images processed inaccordance with an aspect of the disclosed invention;

FIG. 4 is a vectorized representation of the geometry of a clusterscreen used in accordance with the disclosed system and method;

FIGS. 5A and 5B are representative illustrations of watermark images;

FIG. 6 illustrates watermarked images formed in accordance with thedisclosed system and method;

FIGS. 7A and 7B is an example of a public key screen that may be used todetect the watermarks in the images of FIG. 6;

FIGS. 8A-8D are illustrative examples of the resultant retrieval of themark embedded in FIG. 6; and

FIG. 9 is a block diagram depicting an embodiment of the system andrelated methods described herein.

The various embodiments described herein are not intended to limit theinvention to those embodiments described. On the contrary, the intent isto cover all alternatives, modifications, and equivalents as may beincluded within the spirit and scope of the invention as defined by theappended claims.

DETAILED DESCRIPTION

As more particularly set forth below, the disclosed system and methodsare directed to the generation of digital watermarks, includinggenerating separate gray-scale watermarked images using halftonestructures with different Rotation Angles, such that the watermarks maybe retrieved, or viewed, using the same public key overlaid atop theimages at different Orientation Angles.

A smooth phase transition between 0 and π for the region 210 of FIG. 2Acan be achieved by using a three-dimensional threshold array asdisclosed by the above-referenced co-pending applications filedconcurrently herewith: U.S. application Ser. No. ______, Attorney DocketNo. 20051772-US-NP, entitled “______”; and U.S. application Ser. No.______, Attorney Docket No. ______-US-NP, entitled “______” previouslyincorporated herein by reference. The halftone output with the embeddedwatermark pattern looks like the halftone pattern 300 shown in FIG. 3A.The boundary or seam 320 between the phased shifted central region 310and the balance of the image is much less visible than the seam 220 inFIG. 2A. When a reference key, such as the pattern 240 of FIG. 2B isplaced atop the halftone pattern 300 of FIG. 3A, a somewhat blurredsquare, such as depicted in region 360 in FIG. 3B, will be retrieved asthe digital watermark.

Unlike conventional halftoning techniques, the threshold value for aparticular pixel is not chosen from a halftone screen based solely onthe spatial coordinates, x and y, of the pixel, but rather the digitalwatermark embedding method uses a three-dimensional threshold array withan additional dimension specified by the phase shift, or moreparticularly the phase shift relative to an initial zero-shift halftonescreen. During the halftone process, the threshold value for each pixelis chosen from the three-dimensional array (e.g., array 952 in FIG. 9)by specifying the spatial coordinates x and y, as well as a desiredphase shift s. In the following discussion T(x, y, s) is employed torepresent the halftone threshold value as a function of variables x, yand s.

Referring now to FIG. 4, using a vector notation, the geometry of acluster screen can be specified by two spatial vectors, V_(a)(x_(a),y_(a)) and V_(b)(x_(b), y_(b)), as shown at in FIG. 4. As an example, a45 degree, 106 line-per-inch (LPI) cluster screen for a 600 dot-per-inch(DPI) printer can be represented by two vectors, V_(a)(4, 4) andV_(b)(−4, 4).

For halftoning images specified by 8 bits, or gray levels between 0 to255, a common design of the two-dimensional threshold array with a givencluster geometry can be described mathematically as

T(x,y)=128−127·{cos [kπ(x·y _(a) +y·x _(a))]+cos [kπ(x·y _(b) +y·x_(b))]}/2,  (1)

where k is a scaling factor constant.

The equation, sometimes referred to as the dot profile, providesround-dot or round-hole shapes for the halftone outputs in the highlight or the shadow part of an image, and checkerboard-like patterns forthe middle tones. The dot profile T(x, y) in Equation 1 is used as theinitial zero-shift halftone screen, or T(x, y, 0). The three-dimensionalthreshold array, which is also a function of the phase shift used forwatermark embedding, can be obtained by using a slight modification ofEquation 1, and expressed as

T(x,y)=128−127·{cos [kπ(x·y _(a) +y·x _(a))+s]+cos [kπ(x·y _(b) +y·x_(b))+s]}/2,  (2)

where s is the phase shift in radians.

The resolution of the phase shift depends on the application. Ingeneral, a higher resolution provides better watermark hiding butrequires larger memory space to store the three-dimensional array.Practically, for most applications it is possible choose N, the numberof steps for a linear phase transition from zero to π, equal to 255.Therefore, it is possible to interpret the gray-levels in terms ofdesired phase shift. To embed a black/white watermark into halftoneimages, a π shift for all the black areas and no shift for the whitebackground is needed. Consider using 0 for the white and 255 for acomplete black, we may interpret the white, or the gray level 0, as azero phase shift and the complete black, or the gray level 255, as a πphase shift. In other words,

s=g·π/N,  (3)

where g is the gray level, N=255 is the total number of gray levels ands is the phase shift. As will be appreciated, a smooth phase transitionmay be necessary to hide seams caused by the imposition of the watermarkimage.

Referring to FIGS. 5A and 5B, the desired phase transition can berepresented by a blurred watermark image as shown at 560 in FIG. 5B,which may be produced from the original bi-level watermark image shownat 550 in FIG. 5A, wherein all gray levels between 0 and 255 in theblurred image can be interpreted as intermediate steps between phasezero and phase π. The blurring process may be conducted using well-knownlow-pass filtering methods. The proper low-pass filters used in theprocess can be determined in practice by balancing the watermark hidingeffect and the contrast of retrieved watermarks. Experimental resultssuggest that the area of the low-pass filter should be large enough tocover a plurality of clusters, more particularly at least about tenclusters, to provide a satisfactory result.

As described above, in reference to FIG. 4, the geometry of a clusterhalftone structure used for embedding the phase shifted digitalwatermark is described by the two spatial vectors, V_(a)(x_(a), y_(a))and V_(b)(x_(b), y_(b)) which can be referred to as the first halftonestructure V_(a1)(x_(a1), y_(a1)) and V_(b1)(x_(b1), y_(b1)) shown at402. It has further been found, that a plurality of different clusterhalftone structures can be generated by rotating the halftone structuredefined by these spatial vectors to a plurality of different, arbitraryRotation Angles, referred to generally as RA₁-RA_(n) while keeping theinterior angle between the vectors and the vector amplitudes the same.For halftone screens with a 90 degree interior angle (α), and an equalvector amplitude, the rotation angles (RA) may vary between 0 and 90degrees. An example of a second cluster halftone structure described bytwo spatial vectors V_(a2)(x_(a2), y_(a2)) and V_(b2)(x_(b2), y_(b2))that can also be used for embedding the phase shifted digital watermarkis shown at 404 in FIG. 4. This second halftone structure 404 has aRotation Angle of 45 degrees (RA=45°) with respect to the first halftonestructure 402, as shown. The Rotation Angle (RA) is equal to the anglebetween a vector of the first halftone structure, such as V_(a1)(x_(a1),y_(a1)) or V_(b1)(x_(b1), y_(b1)), and the corresponding vector of thesecond halftone structure, V_(a2)(x_(a2), y_(a2)) or V_(b2)(x_(b2),y_(b2)) respectively.

The interior angle the interior angle α₁ between the two vectorsV_(a1)(x_(a1), y_(a1)) and V_(b1)(x_(b1), y_(b1)) of the first halftonestructure 402 is equal to the interior angle α₂ between the two vectorsV_(a2)(x_(a2), y_(a2)) and V_(b2)(x_(b2), y_(b2)) of the second halftonestructure 404; 90 degrees in this example. Also, the vector amplitude|V_(a1)(x_(a1), y_(a1))| of the first vector of the first halftonestructure is equal to the vector amplitude |V_(a2)(x_(a2), y_(a2))| ofthe first vector of the second halftone structure, and the vectoramplitude |V_(b1)(x_(b1), y_(b1))| of the second vector of the firsthalftone structure is equal to the vector amplitude |V_(b2)(x_(a2),y_(b2))| of the second vector of the second halftone structure.

A plurality of these cluster halftone structures, each having adifferent Rotation Angle (RA), can be used for embedding the phaseshifted digital watermark to create a plurality of digital watermarkedimages. A single public key formed from one of the halftone structures,typically one having a Rotation Angle of zero degrees for simplicity,though others can be used, can be used to retrieve the watermark fromall of these watermarked images.

As an example, four of these different halftone structures can each beseparately used to embed a digital watermark pattern into an image asdescribed above, to produce four corresponding digitally watermarkedimages, 600 a-600 d as shown in FIG. 6. The digital watermark is notvisible in the watermarked images 600 a-600 d unless retrieved by anoverlaid public key 700 shown in FIG. 7A. The public key 700 includes astandard checkerboard pattern, shown in close-up at 710 in FIG. 7B,matching the halftone screen used for embedding the watermark. Theoverlaid public key 700 can be oriented at an angle of rotation,referred to as the Orientation Angle (OA), relative to the watermarkedimage. In the examples provided below, a frame 720 bordering the publickey 700 is used for illustrating the Orientation Angle of the key. Theframe 720 includes side portions 720 a and the Orientation Angle isillustrated as the angle between the side portion 720 a and watermarkedimage below. In the examples provided herein, the watermarked image 600a-600 d is shown in a vertical orientation having zero degrees ofrotation. The public key 700 shown in FIG. 7A has an Orientation Angleof zero degrees as shown by the side frame portion 720 a extendingvertically.

By keeping the vector amplitudes |V_(an)(x_(an), y_(an))|,|V_(bn)(x_(bn), y_(bn))| of the corresponding vectors V_(an)(x_(an),y_(an)), V_(bn)(x_(bn), y_(bn)) the same and keeping the interior anglesbetween the vectors the same for each halftone structure used forembedding, and using different Rotation Angles (RA) for each halftonestructure used for embedding, all of the watermarks can be retrievedusing the same public key, also referred to as the single public key700. The single public key 700 is formed using a matching halftonestructure, a checkerboard pattern, described by spatial vectorsV_(a)(x_(a), y_(a)) and V_(b)(x_(b), y_(b)) having the same vectoramplitudes and interior angle. Stated another way, the single public key700 has the same halftone frequencies and angles of the halftonestructure used to embed the watermark image. The watermark embedded ineach image 600 a-600 d is retrieved when the single public key 700 isoverlaid on the digitally watermarked image and rotated to anOrientation Angle matching the Rotation Angle RA of the halftonestructure used for embedding the watermark.

As an example, which should not be considered limiting, by rotating theabove mentioned halftone, a 45 degree, 106 line-per-inch (LPI) clusterscreen for a 600 dot-per-inch (DPI) printer represented by the twovectors, V_(a)(4, 4) and V_(b)(−4, 4), 45, 32, 15 and 0 degrees, the twovectors V_(aRA)(x_(a), y_(a)) and V_(bRA)(x_(b), y_(b)) become:

V_(a45)(5.657, 0) and V_(b45)(0, 5.657),

V_(a30)(1.464, 5.464) and V_(b30)(−5.464, 1.464),

V_(a15)(2.828, 4.899) and V_(b15)(−4.899, 2.828), and

V_(a0)(4, 4) and V_(b0)(−4, 4), respectively keeping the interior angleα and the vector amplitude the same for each.

The digital watermarks can be embedded into the image by selecting anyone of these four configurations for halftoning using thethree-dimensional arrays built as discussed above. In retrievalpractice, the same public key, defined by the vectors V_(a)(4, 4) andV_(b)(−4, 4), can be used for all of them by overlaying the public keytransparency on top of the watermarked image and rotating it relative tothe image until its Orientation Angle matches the Rotation Angle of thehalftone structure used for embedding the watermark image. Creating thepublic key with a halftone structure having a zero Rotation Angleproduces a simple 4×4 checker-board pattern which can be easilyreproduced. However, it should be appreciated that the public key can beproduced using the halftone structure having any Rotation Angle.Referring now to FIGS. 8A-8D, the four images 600 a, 600 b, 600 c and600 d include the same watermark embedded using the different halftonestructures described above.

Watermarked Image 600 a includes the mark embedded using the halftonestructure with R=45 degrees as defined by V_(a45)(5.657, 0) andV_(b45)(0, 5.657). Watermarked Image 600 b includes the mark embeddedusing the halftone structure with RA=30 degrees as defined byV_(a30)(1.464, 5.464) and V_(b30)(−5.464, 1.464). Watermarked Image 600c includes the mark embedded using the halftone structure with RA=15degrees as defined by V_(a15)(2.828, 4.899) and V_(b15)(−4.899, 2.828).Watermarked Image 600 d includes the mark embedded using the halftonestructure with RA=0 degrees as defined by V_(a0)(4, 4) and V_(b0)(−4,4).

The single public key transparency 700, defined by V_(a0)(4, 4) andV_(b0)(−4, 4), is overlaid atop the images 600 a-600 d at variousOrientation Angles relative to the images below, as illustrated by theangle of side portions 720 a of frames 720. In FIG. 8A, the public key700 is overlaid on each image 600 a-600 d at an Orientation Angle of 45degrees which enables only the watermark from image 600 a to beretrieved, since it is embedded with the halftone structure having aRotation Angle of 45 degrees. In FIG. 8B, the public key 700 is overlaidat an Orientation Angle of 30 and only the watermark from image 600 bcan be retrieved, since it is embedded with the halftone structurehaving a Rotation Angle of 30 degrees. In FIG. 8C, the public key 700 isoverlaid at an Orientation Angle of 15 and only the watermark from image600 c can be retrieved, since it is embedded with the halftone structurehaving a Rotation Angle of 15 degrees. In FIG. 8D, the public key 700 isoverlaid at an Orientation Angle of 0 and only the watermark from image600 d can be retrieved, since it is embedded with the halftone structurehaving a Rotation Angle of 0 degrees.

Briefly, the watermark embedding process can be summarized as the seriesof steps generally illustrated in accordance with the block diagram ofFIG. 9. FIG. 9 is an illustration of an exemplary image processingsystem 910, suitable for carrying on digital watermarking of an inputimage. The system 910 includes an image input device, as a source of aninput image 920, such as a scanning device, a computer or imageworkstation for generating images, or a digital camera. The digitalinput image is at least temporarily or partially stored in an imagememory 924. Memory 924, although depicted as a hard disk, may be anysuitable media or installed circuitry including RAM and ROM, removableand permanent and various combinations thereof as are commonly known andused for the storage of digital data such as images. As will be furtherappreciated the memory 924 may be employed merely as a buffer just forthe temporary storage of a portion of the image during processing asdescribed herein.

Similarly, a watermark memory 934 is employed for storing a watermark930 to be embedded in the watermarked output images to be created fromthe stored input image. System 910 further includes an image processor950 or similar control and processing circuitry, such as a digital frontend (DFE) known for use in the processing of digital images forrendering on printing engines and reprographic devices (e.g., Xerox®iGen3™, DocuColor™ and WorkCentre™ systems, etc.). The processor isemployed for embedding a digital watermark 930 into the input image 920to produce digitally watermarked output images 940, wherein thewatermark is embedded into the image using a halftone structure having afirst Rotation Angle to produce a first watermarked image and embeddedinto the image using a halftone structure having a second Rotation Angleto produce a second watermarked image. As will be appreciated the outputimages 640 may be rendered on any image output device such as a suitablemarking or printing engine 990 capable of rendering grayscale output onone or more media.

First, system 910 generates three-dimensional (3D) threshold arrays 952for all configurations with different Rotation Angles as describedabove, and stores the result into a memory. Alternatively, the thresholdvalues for a plurality of given x, y and s values can also be calculatedpixel-wise during the embedding process using Equation 2 above. Storingthe pre-calculated result into the processor memory, in 3D thresholdarray 952 is intended for speeding up the halftone process. Accordingly,it will be appreciated that various halftone result generation andstorage techniques may be employed in other alternative embodiments.

Next, for a given watermark pattern 930, a low-pass filter 954 may beapplied to smooth out edges of the watermark image and the resultantimage is then stored in memory as a multi-bit gray image (e.g., 8-bit),where the different gray levels represent different phase shifts forwatermark embedding. If the original watermark pattern does not containany high-frequency details this step may be omitted, if so desired.

The input image 920 and the processed watermark image obtainedpreviously are then read in by the image processor 950 and a pixel-wisehalftoning operation is conducted. In accordance with the disclosedembodiments, the three-dimensional threshold array 952, stored in memoryaccessible to the processor 950, is employed as an input to a thresholdoperation 956. In response to location coordinates x and y, the inputvalue from the input image, and the threshold value determined by thecoordinates x, y, and the phase shift s given by Equation 3, a resultinggray level g is determined for a plurality of coordinate locations toproduce the processed watermark image 940.

As will be appreciated by those familiar with the design of imageprocessing systems, the image processor 950 further includes timing andcontrol operation block 958, which controls the flow of data andprocessing operations within the image processor, including anybuffering of the image data as depicted in buffers 960 and 962. A widevariety of hardware may be employed to achieve the functionalitydepicted with regard to the image processor, including dedicated imageprocessing chipsets and conventional computer workstations, as well ascombinations thereof or other processing devices.

Once processed, the invisible digital watermark is embedded into theoutput images 940, wherein the watermark is embedded into each image atdifferent Rotation Angles. The watermarked image containing the embeddedinvisible digital watermarks, is then provided as input to the printingengine 990 for rendering. A single public key can be used for retrievingthe watermark from each watermarked image as described above,

It will be appreciated that various of the above-disclosed embodimentsand other features and functions, or alternatives thereof, may bedesirably combined into many other different systems or applications.Also, various presently unforeseen or unanticipated alternatives,modifications, variations or improvements therein may be subsequentlymade by those skilled in the art which are also intended to beencompassed by the following claims.

1. A method creating digital watermarks in grayscale images, comprising:receiving the first image to be watermarked; determining the firstwatermark to be embedded in the first image; embedding the firstwatermark into the first image by halftoning the first image using athree-dimensional threshold array having a phase shift value as an inputand a first halftone structure defined by the spatial vectorsV_(a1)(x_(a1), y_(a1)) and V_(b1)(x_(b1), y_(b1)); outputting a firstwatermarked grayscale image containing the first digital watermarkembedded with the first halftone structure; receiving the second imageto be watermarked; determining the second watermark to be embedded inthe second image; embedding the second watermark into the second imageby halftoning the second image with a three-dimensional threshold arrayhaving a phase shift value as an input and a second halftone structuredefined by the spatial vectors V_(a2)(x_(a2), y_(a2)) and V_(b2)(x_(b2),y_(b2)), wherein the interior angle between the two vectorsV_(a1)(x_(a1), y_(a1)) and V_(b1)(x_(b1), y_(b1)) of the first halftonestructure is equal to the interior angle between the two vectorsV_(a2)(x_(a2), y_(a2)) and V_(b2)(x_(b2), y_(b2)) of the second halftonestructure, the vector amplitude |V_(a1)(x_(a1), y_(a1))| of the firstvector of the first halftone structure is equal to the vector amplitude|V_(a2)(x_(a2), y_(a2))| of the first vector of the second halftonestructure, and the vector amplitude |V_(b1)(x_(b1), y_(b1))| of thesecond vector of the first halftone structure is equal to the vectoramplitude |V_(b2)(x_(a2), y_(b2))| of the second vector of the secondhalftone structure; and outputting a second watermarked grayscale imagecontaining the second watermark embedded with the second halftonestructure.
 2. The method defined in claim 1 further comprising: creatinga public key by using the first halftone structure defined by thespatial vectors V_(a1)(x_(a1), y_(a1)) and V_(b1)(x_(b1), y_(b1)) forretrieving the digital watermark from the first and second image prints.3. The method defined in claim 2 further comprising; retrieving thefirst digital watermark from the first watermarked image using thepublic key without rotation.
 4. The method defined in claim 3 furthercomprising; retrieving the second digital watermark from the secondimage print using the public key oriented with respect to the secondwatermarked image with a Rotation Angle equal to the angle between thefirst vector V_(a1)(x_(a1), y_(a1)) of the first halftone structure andthe first vector V_(a2)(x_(a2), y_(a2)) of the second halftonestructure.
 5. The method defined in claim 1 wherein the embedding stepsfurther comprise conducting pixel-wise haltoning of the image to bewatermarked using the three-dimensional threshold arrayT(x,y)=128−127·{cos [kπ(x·y _(a) +y·x _(a))+s]+cos [kπ(x·y _(b) +y·x_(b))+s]}/2 where s is the phase shift in radians.
 6. The method definedin claim 5 wherein s=g·π/N where g is the gray level of the watermarkand N is the total number of gray levels contained in the watermark. 7.The method defined in claim 1 wherein the spatial vectors V_(a)(x_(a),y_(a)) and V_(b)(x_(b), y_(b)) do not have to be defined by integers. 8.The method defined in claim 1 wherein the spatial vectors represent a 45degree 106 LPI (line-per-inch) cluster screen for a 600 DPI(dot-per-inch) printer.
 9. A method of creating a plurality of grayscaleimages each having a retrievable invisible digital watermark comprising:receiving the grayscale image to be watermarked; determining thegrayscale watermark to be embedded in the image; and creating aplurality of digital watermarked grayscale images by embedding a digitalwatermark into the image by halftoning the image with different halftonestructures each defined by the spatial vectors V_(a)(x_(a), y_(a)) andV_(b)(x_(b), y_(b)) having the same interior angle α, the same vectoramplitudes |V_(a)(x_(a), y_(a))| and |V_(a)(x_(a), y_(a))|, and adifferent Rotation Angle, wherein the watermark can be retrieved fromthe plurality of images using a single public key.
 10. The methoddefined in claim 9 wherein the step of creating a plurality of digitalwatermarked images includes embedding the digital watermark into theimage by halftoning the image using a three-dimensional threshold arrayhaving a phase shift value as an input value.
 11. The method defined inclaim 9 further comprising: creating the single public key defined byspatial vectors V_(a)(x_(a), y_(a)) and V_(b)(x_(b), y_(b)) having theinterior angle α and the same vector amplitudes |V_(a)(x_(a), y_(a))|and |V_(a)(x_(a), y_(a))| as the halftone structures used for embeddingthe digital watermark into the image.
 12. The method defined in claim 11further comprising: retrieving the digital watermark from each of thedigital watermarked images using the single public key.
 13. The methoddefined in claim 12 wherein the retrieving step further comprises:orienting the single public key with respect to the watermarked imagesat different Orientation Angles for retrieving the watermark.
 14. Themethod defined in claim 9 wherein the spatial vectors represent a 45degree 106 LPI (line-per-inch) cluster screen for a 600 DPI(dot-per-inch) printer.
 15. The method defined in claim 14 wherein thehalftone structures are defined by the spatial vectors V_(a45)(5.657, 0)and V_(b45)(0, 5.657) for a Rotation Angle of 45 degrees, V_(a30)(1.464,5.464) and V_(b30)(−5.464, 1.464) for a Rotation Angle of 30 degrees,V_(a15)(2.828, 4.899) and V_(b15)(−4.899, 2.828) for a Rotation Angle of15 degrees, and V_(a0)(4, 4) and V_(b0)(−4, 4) for a Rotation Angle of 0degrees.
 16. A system for digital watermarking of an image, comprising:an input image source; image memory for storing the input image to bewatermarked; watermark memory for storing the watermark to be embeddedin the image; and an image processor including a three-dimensionalthreshold array where at least one input thereto is a phase shift valuefor embedding an invisible digital watermark into the image byhalftoning the image with the three-dimensional threshold array using aplurality of halftone structures each defined by the spatial vectorsV_(a)(x_(a), y_(a)) and V_(b)(x_(b), y_(b)) having the same interiorangle α, the same vector amplitudes |V_(a)(x_(a), y_(a))| and|V_(a)(x_(a), y_(a))|, and a different Rotation Angle to produce aplurality of digitally watermarked output images.
 17. A method forretrieving a watermark image from a plurality of digitally watermarkedgrayscale images, each formed by halftoning an image with a halftonestructure having a different arbitrary Rotation Angle comprising:overlaying a single public key transparency having halftone frequenciesand angles matching the halftone structure used for forming theplurality of digital watermarked grayscale images atop each of thewatermarked images and orienting the key with respect to the watermarkedimages at Orientation Angles matching the Rotation Angles used forembedding the watermark image into the corresponding digitallywatermarked images to retrieve the watermark image.
 18. The methoddefined in claim 17 wherein the public key transparency includes acheckerboard pattern.
 19. The method defined in claim 18 wherein thewatermark image was embedded into the digitally watermarked images ausing a three-dimensional threshold array having phase shift value as aninput.